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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A counterexample in nonlinear interpolation
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by Michael Cwikel PDF
Proc. Amer. Math. Soc. 62 (1977), 62-66 Request permission


If $({A_0},{A_1})$ is an interpolation pair with ${A_0} \subset {A_1}$ and T is a possibly nonlinear operator which maps ${A_0}$ into ${A_0}$ and ${A_1}$ into ${A_1}$ and satisfies ${\left \| {Ta} \right \|_{{A_0}}} \leqslant C{\left \| a \right \|_{{A_0}}}$ and ${\left \| {Tb - Tb’} \right \|_{{A_1}}} \leqslant C{\left \| {b - b’} \right \|_{{A_1}}}$ for all $a \in {A_0}$ and b, $b,b’ \in {A_1}$ and for some constant C, then it is known that T also maps the real interpolation spaces ${({A_0},{A_1})_{\theta ,p}}$ into themselves. We give an example showing that T need not map the complex interpolation spaces ${[{A_0},{A_1}]_\theta }$ into themselves. It is also seen that quasilinear operators may fail to preserve complex interpolation spaces.
  • A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
  • Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
  • P. Krée, Interpolation d’espaces vectoriels qui ne sont ni normés, ni complets. Applications, Ann. Inst. Fourier (Grenoble) 17 (1967), no. fasc. 2, 137–174 (1968) (French). MR 229026
  • J.-L. Lions, Some remarks on variational inequalities, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) Univ. Tokyo Press, Tokyo, 1970, pp. 269–282. MR 0267420
  • Jaak Peetre, Interpolation of Lipschitz operators and metric spaces, Mathematica (Cluj) 12(35) (1970), 325–334. MR 482280
  • Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. I. Principal properties, J. Math. Mech. 13 (1964), 407–479. MR 0163159
  • L. Tartar, Interpolation non linéaire et régularité, J. Functional Analysis 9 (1972), 469–489 (French). MR 0310619, DOI 10.1016/0022-1236(72)90022-5
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 62-66
  • MSC: Primary 46M35; Secondary 47H99
  • DOI:
  • MathSciNet review: 0433227